Signal reconstruction in compressive sensing involves finding a sparse solution that satisfies a set of linear constraints. Several approaches to this problem have been considered in existing reconstruction algorithms. They each provide a trade-off between reconstruction capabilities and required computation time. In an attempt to push the limits for this trade-off, we consider a smoothed l0 norm (SL0) algorithm in a noiseless setup. We argue that using a set of carefully chosen parameters in our proposed adaptive SL0 algorithm may result in significantly better reconstruction capabilities in terms of phase transition while retaining the same required computation time as existing SL0 algorithms. A large set of simulations further support this claim. Simulations even reveal that the theoretical l1 curve may be surpassed in major parts of the phase space.